Evolution Mixed Variational Inclusions with Optimal Control

Evolution mixed maximal monotone variational inclusions with optimal control, in reflexive Banach spaces, are analized. Solvability analysis is performed on the basis of composition duality principles. Applications to nonlinear diffusion constrained problems, as well as to quasistatic elastoviscoplastic contact problems exemplify the theory.


Introduction
The aim of this paper is to study primal and dual evolution mixed maximal monotone inclusions with optimal control, in a variational setting of reflexive Banach spaces.Distributed as well as boundary state constraints are considered through intrinsic control subdifferential mechanisms.For a mixed variational structure of the governing state model, and for computational purposes, imposed constraints are handled via compositional dualization.The state system solvability is analized on the basis of composition duality principles in the sense of (Alduncin, G., 2007a(Alduncin, G., , 2007b(Alduncin, G., , 2011b)), which lead to a primal-dual variational analysis.
The solvability analysis is performed by adapting the study of (Akagi, G. & Ôtani, M., 2004), to our primal and dual evolution mixed maximal monotone subdifferential systems.The optimality analysis is given on the basis of Migórski's work on optimal control of evolution hemivariational inequalities (Migórski, S., 2001).
Here, in accordance with (Alduncin, G., 2007a(Alduncin, G., , 2007b(Alduncin, G., , 2011b)), we consider primal as well as dual evolution mixed variational inclusions, since both formulations are of relevant importance in computational mechanics.Through the sequel, applications to nonlinear diffusion constrained processes, and quasistatic elastoviscoplastic variational contact problems, exemplify the control theory.

Evolution Mixed Variational Inclusions
In this section, we begin introducing the evolution mixed inclusion state problems of the theory.Here, we follow our previous work on evolution mixed variational inclusions (Alduncin, G., 2007b), in an abstract sense of constrained initial boundary value problems.
As a stationary functional framework for the dynamical systems of the theory, we consider primal and dual Ω-field reflexive Banach spaces V(Ω) and Y * (Ω), with respective topological duals denoted by V * (Ω) and Y(Ω), relative to a spatial bounded domain Ω ⊂ ℜ d , d ∈ {1, 2, 3}.Denoting by ∂Ω the boundary of the domain Ω, assumed to be Lipschitz continuous, let π be the corresponding linear continuous primal trace operator, defined in V(Ω), with values in a reflexive Banach boundary space B(∂Ω), with topological dual B * (∂Ω), and satisfying the compatibility condition (C π ) π ∈ L(V(Ω), B(∂Ω)) is surjective.
Here, ∂F : V → 2 V * stands for the evolution primal subdifferential of the system model, ∂G * : Y * → 2 Y corresponds to the evolution subdifferential control mechanism for the distributed constraints of the problem, and ∂Ψ : B → 2 B * is a maximal monotone subdifferential modeling boundary conditions as well as boundary constraints.Also, π T ∈ L(B * , V * ) denotes the transpose of the evolution primal trace operator π ∈ L(V, B).  (Ekeland, I. & Temam, R.,1974).Thereby, the boundary condition and constraint term of the primal equation of mixed problem ( M) specializes into the specific form Hence, on the basis of primal boundary compatibility condition (C π ), equivalent to the lower boundedness of its transpose operator π T ∈ L(B * (∂Ω), V * (Ω)) (Yosida, K., 1974), the following result is concluded via the injectivity of π T and compositional dualization (Alduncin, G., 2010).
Lemma 1 Under compatibility condition (C π ), the primal boundary term of (1) is such that where, for any functional v * ∈ R(π T P ) ⊂ V * , c * ∈ B * ∂Ω P is its π T P -preimage: v * = π T P c * .Therefore, taking into account compatibility condition (C π ), the boundary conditions and constraint of the problem, can be incorporated variationally to primal evolution mixed problem ( M) as follows. (M) This subdifferential form of the evolution mixed problem will be appropriate for performing its qualitative analysis.
Example 2 In order to illustrate the primal theory of evolution mixed variational inclusions, we consider a distributed constrained nonlinear diffusion problem, modeled by the mixed state system of conservation, constitutive and distributed constraint equations Here, u denotes the diffusive field of a transport process evolving in Q = Ω × (0, T ) (for instance of mass concentration or temperature), with linear flux vector field w.Also, b * corresponds to the divergence of the nonlinear flux vector field denoted by w, and p * is the distributed control source field governed by a maximal monotone control mechanism ∂ϕ * (Duvaut, G. & Lions, J.-L., 1972).For the nonlinear diffusion constitutive equation, we consider the potential and subdifferential the latter in the Gâteaux sense (Vainberg, M. M., 1973).Notice that by convex dualization the nonlinear constitutive equation of the state system (3) is equivalently expressed by b * ∈ ∂φ(u), φ * denoting the conjugate of φ, and by definition b * = div w.Hence, the nonlinear flux vector field of the model turns out to be w = −∥grad u∥ p−2 grad u, and the total flux vector field of diffusion is given by the sum w + w.
Thereby, an appropriate functional framework of a primal Ω-field space V(Ω), of u-diffusive functions, and a dual Ω-field space Y * (Ω), of w-linear flux vector, b * -nonlinear flux divergence and p * -distributed source control functions, is provided by the reflexive Banach Sobolev spaces (Adams, R. A., 1975).
In this dual case, a mechanical interpretation of problem ( M * ) is that now the V * -primal variational equation corresponds to a variational model of the imposed interior or distributed constraints, to be dualized for computational purposes, with an incorporated boundary term upon the application of its Green formula, while the Y(Ω)-dual subdifferential equation corresponds to the constitutive or balance equation of the system.On the other hand, the essential and natural boundary conditions, as well as the boundary constraints of the problem, are to be variationally modeled as for the primal case of the previous subsection.Hence, the dual evolution mixed problem ( M * ), with boundary conditions primally incorporated and boundary control dualized, takes the subdifferential form p * (0) = p * 0 , which will be appropriate for qualitative analysis.
Example 3 For the illustration of the dual theory of evolution mixed variational inclusions, we consider a boundary constrained quasistatic evolution elastoviscoplastic problem.Let the spatial domain Ω be occupied by a solid body in its reference configuration, undergoing small strains and displacements, and whose material is characterized by the general elastoviscoplastic constitutive model of a nonlinear Maxwell viscoelastic type (Le Tallec, P., 1990).Here, S denotes the Cauchy stress tensor field, w = u stands for the velocity field, C is the elasticity tensor, symmetric and positive definite, with inverse C −1 , the elastic compliance tensor.Further, ∂φ standing for the maximal monotone dissipation subdifferential, its dual ∂φ * corresponds to the subdifferential of a conjugate dissipation superpotential φ * or yield functional, and ∇ s w ≡ 1/2(∇w + ∇w T ) is the total small strain rate tensor field, induced by the velocity w.To this stress evolution constitutive equation, an initial stress value S(0) = S 0 is assigned in the domain Ω.
As a balance equation for the system, we have the quasistatic equilibrium equation b denoting the body force data.With respect to the constraints of the problem, we consider on disjoint complementary sub-boundaries of domain Ω, ∂Ω = ∂Ω C ∪ ∂Ω DN , of a Lipschitz continuous type, a bilateral contact model of interaction with a rigid fixed foundation on ∂Ω C × (0, T ), under the nonseparation condition w ν = 0, and Tresca's law of dry friction with prescribed shear bound function f * > 0.Then, we associate with equilibrium equation ( 9) the bilateral contact constraint, with given friction, subdifferentially modeled by the dual inclusions Thereby, from model equations ( 8)-( 11), the dual quasistatic mixed elastoviscoplastic contact physical problem to be considered is the following, for given data (b, f * , w, s, S 0 ).

Composition Duality Principles
For the existence analysis of primal and dual evolution mixed state systems (M) and (M * ), we establish in this section composition duality principles that lead to corresponding primal and dual evolution state inclusions equivalent in a solvability sense.Toward this end, we shall follow the duality approaches of our resent paper (Alduncin, G., 2011b).

Primal evolution duality principles
For the primal evolution mixed state system (M), we first establish a direct duality principle, following (Le Tallec, P., 1990), in terms of its primal admissibility set, to which any primal component solution belongs, for a.e.t ∈ (0, T ), Then, introducing the coupling compatibility condition ), the next dual compatibility condition of the problem can be concluded (Alduncin, G., 2011b).
) be fulfilled.Then the dual admissibility set of primal evolution mixed problem (M) is characterized by the subdifferential of the indicator functional I S (p,λ) (t) .
We note that due to condition (C Λ T ,π T C ) and the Close Range Theorem, the sum of the ranges of the transpose coupling operators coincide with the sum of the polar kernels of the own coupling operators; i.e., R(Λ Thereby, the following primal evolution duality principle for problem (M) is readily concluded.
Theorem 5 Under compatibility condition (C Λ T ,π T C ), primal evolution mixed problem (M) is solvable if, and only if, the primal evolution problem For a complementary primal duality principle to evolution mixed state inclusion (M), we next apply the composition duality methodology established in (Alduncin, G., 2007b).Toward this end, we first give a compositional interpretations of dual admissibility characterization (17), which is based on the following compositional result (Alduncin, G., 2011b).

Lemma 6 The dual inclusions of problem (M) necessarily satisfy the compositional dualizations
where v g ∈ V is a fixed Λ-preimage of function g: Λv g = g.
Hence, from Lemma 6 and Theorem 5, and introducing the primal operator and right-hand side term we can conclude the next necessary primal evolution composition duality principle.
Theorem 7 Primal evolution mixed problem (M) is solvable, only if the primal evolution problem

is a solution of mixed problem (M) then component u is a primal admissible solution of problem ( P). Moreover, compositional primal problem ( P) is solvable if primal evolution problem (P) is solvable.
Further, following our previous study (Alduncin, G., 2007b), we introduce the classical primal evolution compatibility conditions which lead to the compositional operator equalities (Ekeland, I. & Temam, R., 1974) Hence, by dualization of the dual inclusions of problem (M), and the validity of ( 18) in its equivalence sense, due to composition duality relations (20), Theorem 7 applies then, additionally, in a sufficiency sense (Alduncin, G., 2007b; Theorem 2.2).That is, the following alternative primal evolution duality principle holds true. ) and (C (G,Λ),(Ψ C ,π C ) ) we have the following (without boundary constraint): This first condition is classically satisfied (Girault, V. & Raviart, P.-A., 1986), and the latter may be consider as a proper condition for control mechanisms (Duvaut, G. & Lions, J.-L., 1972).Then, under condition (C (G,Λ) cd ), in accordance with Theorem 8, primal evolution mixed problem (M cd ) is solvable if, and only if, the primal evolution variational problem is solvable.Moreover, due to property (C (Λ T ) cd ), compositional primal problem ( P cd ) has as an interpretation primal problem (P cd ) of Theorem 5, with primal admissibility set (16) characterized by its indicator subdifferential (17).

Dual evolution duality principles
Considering now the dual evolution mixed state system (M * ), we first establish as in the previous primal subsection a direct duality principle.In this dual case, we dualized compositionally its dual subdifferential boundary equation, based on primal trace compatibility condition (C π ) (Alduncin, G., 2005a); i.e., Hence, the primal equation of system (M * ) becomes where primal definition ( 19) is in force.Consequently, the corresponding dual admissibility set of the system, to which any dual component solution belongs, for a.e.t ∈ (0, T ), turns out to be Then, introducing the coupling compatibility condition the next primal compatibility condition of the problem can be achieved (Alduncin, G., 2011b).
Lemma 4 * Let compatibility condition (C −Λ ) be satisfied.Then the primal admissibility set of dual evolution mixed problem (M * ) is characterized by the subdifferential of the indicator functional I S v * (t) .
Therefore, the following dual evolution duality principle for problem (M * ) is obtained.
Theorem 5 * Under compatibility condition (C −Λ ), dual evolution mixed problem (M * ) is solvable if, and only if, the dual evolution problem As in the primal case, for a complementary dual duality principle to evolution mixed state inclusion (M * ), we apply the composition duality methodology of (Alduncin, G., 2007b).First, we give a compositional interpretation of primal admissibility characterization (25), based on the next compositional result (Alduncin, G., 2011b).
Lemma 6 * The primal inclusion of problem (M * ) necessarily satisfies the compositional dualization Thereby, the following necessary dual evolution composition duality principle holds true.
Theorem 7 * Dual evolution mixed problem (M * ) is solvable, only if the dual evolution problem Furthermore, following (Alduncin, G., 2007b), introducing the classical dual evolution compatibility condition the compositional operator equality (Ekeland, I. & Temam, R., 1974) holds true.Thus, by dualization of the primal inclusion of problem (M * ), ( 23), and the validity of ( 26), but in its equivalence sense due to compositional dualization ( 27), Theorem 7 * applies now in an equivalence sense (Alduncin, G., 2007b).Therefore, the following alternative dual evolution duality principle is achieved.Example 10 In order to relate dual evolution mixed quasistatic contact problem ( M * cp ) of Example 3 with the dual evolution mixed state system of the theory, (M * ), we will have to dualize its contact subdifferential equations.To this end, taking into account primal trace compatibility condition (C1 π cp ), by compositional dualization we have the duality result (Alduncin, G., 2005a) γ which leads to the dual evolution mixed equivalent version Here, for a.e.t ∈ (0, T ), the primal subdifferential ∂ F : V(Ω) → 2 V * (Ω) is given by where the subdifferential sum rule has been used (Ernst, E. & Théra, M., 2009), and the right-hand side term b . Thereby, for this particular dual evolution mixed state system, the corresponding dual duality principle follows, in accordance with the statically admissible set of the mechanical problem, for a.e.t ∈ (0, T ), its total small strain compatibility property (Girault, V. & Raviart, P.-A., 1986) and its kinematic compatibility characterization the subdifferential of the indicator functional I S w * (t) .Indeed, the dual evolution duality principle for problem (M * cp ) is given as follows (cf.Theorem 5 * ): dual evolution mixed contact problem (M * cp ), equivalent to ( M * c p ), is solvable if, and only if, the dual evolution problem then there is a kinematically admissible velocity field w ∈ V, with −Hw ∈ ∂I S w * (S), such that (w, S) is a solution of mixed problem (M * cp ) and, conversely, if (w, S) ∈ V × X is a solution of mixed problem (M * c p ) then statically admissible stress field S is a solution of dual problem (D c p ).For a compositional interpretation of kinematic compatibility condition (30), we apply the necessary duality result (cf.Lemma 6 * ), related to the primal velocity inclusion of mixed problem (M * cp ), where Therefore, in accordance with Theorem 7 * , a necessary dual evolution composition duality principle for the elastoviscoplastic contact problem is concluded: dual evolution mixed problem (M * cp ) is solvable only if the dual evolution problem We should note that for a fully dual compositional duality principle in the sense of (Alduncin, G., 2005a(Alduncin, G., , 2005b)), the surjectivity of the total small strain transpose operator −H T ∈ L(Y(Ω), V * (Ω)) would be required, guaranteeing the sufficiency of compositional dualization (32) and, consequently, the sufficiency of the above dual duality principle.However, such a total small strain transpose operator surjectivity would require of a primal quotient space V(Ω) setting (Girault, V. & Raviart, P.-A., 1986).On the other hand, for the validity of the compositional duality principle established by Theorem 8 * , the corresponding classical compatibility condition should be satisfied in order to apply corresponding compositional operator equality (27).

Variational Problems with Optimal Control
In this last section, we state an optimal control problem, governed by primal and dual evolution mixed variational systems (M) and (M * ), performing the state existence analysis on the basis of (Akagi, G. & Ôtani, M., 2004) work on evolution inclusions.The corresponding optimal control existence is given in accordance with (Migórski, S., 2001) study on optimal control of evolution hemivariational inequalities.
Let C stand for the space of controls, a reflexive Banach space, and let K be a closed convex subset of admissible controls.As an optimal control problem, we consider the following ∂Ω C are corresponding κ-optimal states of the primal or dual evolution mixed systems, respectively.In general, the cost functional J : C × (V × Y * × B * ∂Ω C ) → ℜ ∪ {+∞} will be of the form ℓ a non-decreasing real function, be fulfilled.Then primal evolution problem ( P κ ) has a unique solution.
Therefore, collecting results, we conclude the following mixed solvability result.
Example 13 Returning to the distributed constrained nonlinear diffusion problem treated in Examples 2 and 9, we have the primal evolution control problem whose diffusion potential and differential are such that, for v ∈ W 1,p 0 (Ω), Notice that the above differential sum rule holds true since the functional v → ∥ grad v∥ q L q , for q ≥ 2, is of class C 1 .Moreover, it is well known that the nonlinear diffusion variational operator ∂φ = grad T (1 + ∥ grad v∥ p−2 )grad : W 1,p 0 (Ω) → W −1,p * (Ω), is a Lipschitz continuous bounded, strongly monotone and coercive operator (Lions, J.-L., 1969).Therefore, since then primal subdifferential sum (35) holds true in this case due to the proper condition D(ϕ) ∅, of admissible control mechanisms for distributed constraints (with proper lower semicontinuous convex superpotentials ϕ's), assuming the fulfillment of corresponding conditions (C1 φ ) and (C2 φ ), the variational existence results of Theorems 11 and 12 apply to the primal and corresponding mixed evolution distributed constrained nonlinear diffusion problems ( P cd,κ ) and

Dual evolution mixed control problem
Proceeding next with the dual control theory, the dual evolution mixed variational inclusion, governing optimal control problem (O), is given by Here Next, in order to apply (Akagi, G. & Ôtani, M., 2004) existence theorem in a dual sense, we express dual problem ( D κ ), as for the primal case, in a classical subdifferential form.Thus, we introduce the composition dual superpotential where q * f * is the fixed −Λ T -preimage of Lemma 6 * , with effective domain and subdifferential D( , respectively.Also, we assume the corresponding Moreau-Rockafellar-Robinson condition (Ernst, E. & Théra, M., 2009) which guarantees the validity of the dual subdifferential sum rule Then, dual evolution problem ( D κ ) has the subdifferential form and in accordance with (Akagi, G. & Ôtani, M., 2004), the following existence theorem applies to the dual control case.
Theorem 11 * Let the coercivity and boundedness dual conditions ℓ * a non-decreasing real function, be satisfied.Then dual evolution problem ( D κ ) possesses a unique solution.

Evolution optimal control problem
We finally state the solvability of the optimization problem (O), governed by primal and dual evolution mixed state systems (M κ ) and (M * κ ), analyzed in the previous subsections.Considering the evolution hemivariational control analysis of (Migórski, S., 2001), in its monotone variational sense, let the cost or objective functional of optimization problem (O), with general mixed form ( 33 Then, in accordance with (Migórski, S., 2001; Theorem 2), the optimization solvability of the control problem is achieved.

From a mechanical
point of view, the V * -primal variational equation of problem ( M) corresponds to the constitutive or balance equation of the system, with an incorporated boundary term upon the application of its Green formula, while the Y(Ω)-dual subdifferential equation corresponds to a variational model of the imposed interior or distributed constraints, dualized for computational purposes.On the other hand, the essential or primal boundary condition of the problem is modeled variationally by ∂Ψ P (π P u) = ∂I { u} (π P u), and the natural or dual boundary condition by ∂Ψ D (π D u) = { s * }.Here, I { u} denotes the indicator functional imposing the primal boundary value u ∈ B ∂Ω P on the primal part of the boundary ∂Ω P ⊂ ∂Ω, and s * ∈ B * ∂Ω D is the dual boundary value prescribed on the dual part of the boundary ∂Ω D ⊂ (∂Ω\∂Ω P ).Moreover, we consider a third disjoint and complementary part of the boundary, ∂Ω C = (∂Ω\∂Ω P )\∂Ω D , where the boundary constraints of the problem are assigned, and the corresponding boundary control λ * ∈ B * ∂Ω C , to be determined, is variationally modeled by the duality relation λ where, ν denotes the unit outward boundary normal vector, w ν = w•ν and w τ = (I − ν ⊗ ν)w are the normal and tangential components of velocity, and s ν = −Sν•ν and s τ = −(I − ν ⊗ ν)Sν are the contact pressure and the tangential negative traction.Further, as a generalized maximal monotone velocity-traction boundary condition on the Dirichlet-Neumann boundary ∂Ω DN × (0, T ), we impose to the problem the subdifferential equation −Sν ∈ ∂ψ DN (w).(11)In particular, we shall consider the Dirichlet and Neumann boundary conditions of prescribed velocity w, on ∂Ω D ×(0, T ) ∅ (nonempty for convenience), and traction s, on the complementary sub-boundary ∂Ω N × (0, T ), ∂Ω N = ∂Ω DN \∂Ω D .
a solution of primal problem (P) then there is an admissible dual function (p * , λ * ) ∈ Y * × B * ∂Ω C , with Λ T p * + π T C λ * ∈ ∂I S (p,λ) (u), such that u and (p * , λ * ) is a solution of mixed problem (M) and, conversely, if u ∈ W and (p * , λ * ) ∈ Y * × B * ∂Ω C is a solution of mixed problem (M) then primal admissible function u is a solution of primal problem (P).
such that u and (p * , λ * ) is a solution of mixed problem (M * ) and, conversely, if u ∈ V and (p * , λ * ) ∈ X * × B * ∂Ω C is a solution of mixed problem (M * ) then dual admissible function p * is a solution of dual problem (D).

Theorem 8 *
Let compatibility condition (C F * ,−Λ T ) be fulfilled.Then dual evolution mixed problem (M * ) is solvable if, and only if, dual evolution problem ( D) is solvable.Moreover, under condition (C −Λ ), compositional dual problem ( D) is equivalent to dual evolution problem (D).
a solution of problem (M * c p ) then dual field S is a statically admissible solution of problem ( D cp ).Moreover, compositional dual problem ( D cp ) is solvable if dual evolution problem (D cp ) is solvable.